Pricing Defaultable Bonds in a Markov Modulated Market

This article presents a structural model of default risk under macroeconomic conditions. The macroeconomic conditions are assumed to be a finite state of a Markov chain. The innovation of the author’s model is to characterize the firm default, the default-free pure discount bond price, the defaultable bond price, and the credit spread associated with macroeconomic conditions. By using the Wiener–Hopf factorization, the author shows that there is a risk-neutral measure under regime switching. Under macroeconomic conditions, he derives both the default-free bond price and the price of the defaultable bond governed by fundamental systems of partial differential equations. The defaultable yield to maturity, credit spread, and duration are related to the finite state of the Markov chain.

Modelling the Evolution of Credit Spreads using the Cox Process within the HJM Framework: A CDS Option Pricing Model

A four‐factor model (the extended model of Schmid and Zagst) is presented for pricing credit risk related instruments such as defaultable bonds or credit derivatives. It is an advancement of an earlier three‐factor model. In addition to a firm‐specific credit risk factor, a new systematic risk factor in the form of GDP growth rate is included. This new model is set in the context of other hybrid defaultable bond pricing models and empirically compared to specific representatives. We find that a model based only on firm‐specific variables is unable to capture changes in credit spreads completely. However, it is shown that in this model, market variables such as GDP growth rates, non‐defaultable interest rates and firm‐specific variables together significantly influence credit spread levels and changes.

Markovian Defaultable HJM Term Structure Models with Unspanned Stochastic Volatility

We present a pricing kernel that summarizes well the main features of the dynamics of interest rates and risk in postwar U.S. data and use it to uncover how the pricing kernel has moved with the short rate in this data. Our findings imply that standard monetary models miss an essential link between the central bank instrument and the economic activity that monetary policy is intended to affect, and thus we call for a new approach to monetary policy analysis. We sketch a new approach using an economic model based on our pricing kernel. The model incorporates the key relationships between policy and risk movements in an unconventional way: the central bank's policy changes are viewed as primarily intended to compensate for exogenous business cycle fluctuations in risk that threaten to push inflation off target. This model, while an improvement on standard models, is considered just a starting point for their revision. It leads to critical questions that researchers need to answer as they continue to revise their approach to monetary policy analysis. Copyright © 2009 by the National Bureau of Economic Research.

We develop a structural credit model to examine how interactions between default and liquidity affect corporate bond pricing. The model features debt rollover and bond-price-dependent holding costs. Over the business cycle and in the cross-section, the model matches average default rates and credit spreads in the data, and captures variations in bid-ask and bond-CDS spreads. A structural decomposition reveals that default-liquidity interactions can account for 10%–24% of the level of credit spreads and 16%–46% of the changes in spreads over the business cycle. Further, liquidity-related corporate bond financing costs amount to 6% of the total issuance amount from 1996 to 2015. Received July 12, 2015; editorial decision April 15, 2017 by Editor Andrew Karolyi. Authors have furnished an Internet Appendix, which is available on the Oxford University Press website next to the link to the final published paper online.

We introduce the entropic measure transform (EMT) problem for a general process and prove the existence of a unique optimal measure characterizing the solution. The density process of the optimal measure is characterized using a semimartingale BSDE under general conditions. The EMT is used to reinterpret the conditional entropic risk‐measure and to obtain a convenient formula for the conditional expectation of a process that admits an affine representation under a related measure. The EMT is then used to provide a new characterization of defaultable bond prices, forward prices and futures prices when a jump‐diffusion drives the asset. The characterization of these pricing problems in terms of the EMT provides economic interpretations as maximizing the returns subject to a penalty for removing financial risk as expressed through the aggregate relative entropy. The EMT is shown to extend the optimal stochastic control characterization of default‐free bond prices of Gombani & Runggaldier (2013). These methods are illustrated numerically with an example in the defaultable bond setting. The Canadian Journal of Statistics 48: 97–129; 2020 © 2020 Statistical Society of Canada

Merton model has provided a classic theoretical framework for explaining credit spreads. This paper extends Merton model by introducing morphology factor of asset value volatility in the model, and conducts empirical studies on the effect of asset volatility morphology on credit spreads in China’s bond market. The results show that asset volatility morphology is economically important and can explain credit spreads well. Furthermore, this paper analyzes the asymmetric influences of monetary policy on credit spreads and asset volatility morphology. This paper points out that the responses of credit spreads and asset volatility morphology to monetary policy are consistent in the tight liquidity environments. To this end, monetary policy and liquidity, which are two factors that have been ignored by classic Merton model but proved to have significant influences on credit spreads, play roles in influencing credit spreads by changing volatility morphology of asset value. Since asset volatility morphology can reflect the change of investors’ expectation on the default probability of asset, the argument mentioned in the credit spread puzzle that the fundamentals related to bond default probability cannot explain credit spreads needs to be reexamined.

This paper presents a unifying theory for valuing contingent claims under a stochastic term of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processeE for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results. IN RELATION TO the term of interest rates, arbitrage pricing theory has two purposes. The first, is to price all zero coupon (default free) bonds of varying maturities from a finite number of economic fundamentals, called state variables. The second, is to price all interest rate sensitive contingent claims, taking as given the prices of the zero coupon bonds. This paper presents a general theory and a unifying framework for understanding arbitrage pricing theory in this context, of which all existing arbitrage pricing models are special cases (in particular, Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), Ball and Torous (1983), Ho and Lee (1986), Schaefer and Schwartz (1987), and Artzner and Delbaen (1988)). The primary contribution of this paper, however, is a new methodology for solving the second problem, i.e., the pricing of interest rate sensitive contingent claims given the prices of all zero coupon bonds. The methodology is new because (i) it imposes its stochastic directly on the evolution of the forward rate curve, (ii) it does not require an inversion of the term structure to eliminate the market prices of risk from contingent claim values, and (iii) it has a stochastic spot rate process with multiple stochastic factors influencing the term structure. The model can be used to consistently price (and hedge) all contingent claims (American or European) on the term structure, and it is derived from necessary and (more importantly) sufficient conditions for the absence of arbitrage. The arbitrage pricing models of Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), and Artzner and Delbaen (1988) all require an IFormerly titled Bond Pricing and the Term Structure of Interest Rates: A New Methodology.

this paper we present a tree model for defaultable bond prices which can be used for the pricing of credit derivatives. The model is based upon the two-factor Hull-White (1994) model for default-free interest rates, where one of the factors is taken to be the credit spread of the defaultable bond prices. As opposed to the tree model of Jarrow and Turnbull (1992), the dynamics of default-free interest rates and credit spreads in this model can have any desired degree of correlation, and the model can be fitted to any given term structures of default-free and defaultable bond prices, and to the term structures of the respective volatilities. Furthermore the model can accommodate several alternative models of default recovery, including the fractional recovery model of Duffie and Singleton (1994) and recovery in terms of equivalent default-free bonds (see e.g. Lando (1998)). Although based on a Gaussian setup, the approach can easily be extended to non-Gaussian processes that avoid negative interest-rates or credit spreads

The problem of hedgeability and replicatability of European type contingent claims are discussed in an incomplete market with the wealth and the portfolio possibly being constrained. Such problems can be regarded as controllability problem for stochastic systems. For the case of no constraint, using the idea of the Four Step Scheme (Protter and Yong, 1994), we obtain the replicatability of a class of contingent claims (including European call and put options) without assuming ad hoc technical conditions. For the case with wealth and portfolio being constrained, several positive and negative results concerning the hedgeability and replicatability are presented.

The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The uncertainty is specified by a family of martingale probability measures which may not be dominated. We obtain a partial characterization result and a full characterization which extends Avellaneda, Levy and Paras results in the UVM case.

This paper examines optimal consumption and investment choices and the cost of hedging contingent claims in the presence of margin requirements or, more generally, of nonlinear wealth dynamics and constraints on the portfolio policies. Existence of optimal policies is established using martingale and duality techniques under general assumptions on the securities' price process and the investor's preferences. As an illustration, explicit solutions are provided for an agent with ‘logarithmic’ utility. A PDE characterization of the cost of hedging a nonnegative path‐independent European contingent claim is also provided.

In this paper, we study the problem of finding the minimal initial capital (i.e. super-replication value) needed in order to hedge (without risk) European contingent claims in a Markov setting under proportional transaction costs. The main result is that the cheapest (trivial) buy-and-hold strategy is optimal. Such a negative result has been derived previously in different contexts. First, we focus on discrete-time binomial models. We prove that the continuous-time limit of the super-replication value is the cost of the cheapest buy-and-hold strategy. Then, the result is proved in a multivariate continuous-time model with Brownian filtration. As a direct consequence, we obtain an explicit characterization of the hedging set, i.e. the set of all initial positions in the market assets from which the contingent claim can be hedged through some admissible portfolio strategy.

In the following, we develop a model for pricing defaultable debt that is capable of reflecting real world phenomena realistically. Before we actually introduce the model, we make some heuristic observations to motivate the theory. Throughout the following sections we examine and extend these considerations further: Since defaults actually happen, a bond pricing model must be able to deal with default events: For example, in August 1999 Ecuador defaulted on interest rate payments of Brady bonds with a total amount of about US$ 6 billions. In October 1999 there was a second default of US$ 500 millions. Figure 6.1 shows the yields to maturity and S&P ratings of an Ecuadorian sovereign bond that was downgraded to the rating D in September 1999. Rating agencies can’t check the ratings on a daily basis or even continuously. Ratings of rating agencies can only be described by discrete time processes. But the quality of an obligor usually changes according to a continuous time process (see, e.g., figure 6.1). Thus, the quality or the distance to default of an obligor should be observed continuously. Very often the quality of obligors shows mean reversion properties. In case of a downgrade the management of a firm usually takes actions to improve the quality again. Figure 6.2 shows this phenomenon for the firm Hertz. The mean reversion level of the rating of Hertz is A- and A3 for S&P’s and Moody’s rating, respectively. Many models for pricing defaultable zero coupon bonds don’t allow for positive short term credit spreads: As time to maturity tends to zero the model credit spreads tend to zero, too. Obviously, these models are far away from describing reality. In figure 6.3 we show observed corporate credit spreads for different maturities. Even bonds with a rating of A and time to maturity of seven days can have credit spreads which are higher than 500 basis points. Hence, a model must be capable of generating high credit spreads even for short term debt. Yields and credit spreads usually depend on the quality of the obligor. The better the quality the smaller the yields (compare to figure 6.4). The time series of the yields show high correlations between the rating classes A and AAA. Gains and losses are — at least to some extend — caused by changes in the quality of the obligor. Credit spread processes often show mean reversion properties. See, e.g., figure 6.5. Non-defaultable interest rates are usually mean reverting. See, e.g., figure 6.6. KeywordsDefault RiskCredit SpreadBond PriceShort RateCredit DerivativeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.